91Ó°ÊÓ

Sketch a graph of a function whose derivative is always positive.

Find equations for the tangent line and normal line to the circle at the given points. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, tangent line, and normal line. $$ \begin{aligned} &x^{2}+y^{2}=25 \\ &(4,3),(-3,4) \end{aligned} $$

Show that the normal line at any point on the circle \(x^{2}+y^{2}=r^{2}\) passes through the origin.

Use a computer algebra system to differentiate the function. \(g(\theta)=\frac{\theta}{1-\sin \theta}\)

Find the points at which the graph of the equation has a vertical or horizontal tangent line. $$ 25 x^{2}+16 y^{2}+200 x-160 y+400=0 $$

Find the points at which the graph of the equation has a vertical or horizontal tangent line. $$ 4 x^{2}+y^{2}-8 x+4 y+4=0 $$

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=x+\sin x, \quad 0 \leq x<2 \pi\)

Verify that the two families of curves are orthogonal where \(C\) and \(K\) are real numbers. Use a graphing utility to graph the two families for two values of \(C\) and two values of \(K\). $$ x^{2}+y^{2}=C^{2}, \quad y=K x $$

Sketch the graph of a function \(f\) such that \(f^{\prime}>0\) for all \(x\) and the rate of change of the function is decreasing.

The relationship between \(f\) and \(g\) is given. Explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). \(g(x)=-5 f(x)\)